# American Institute of Mathematical Sciences

• Previous Article
H2-stability of some second order fully discrete schemes for the Navier-Stokes equations
• DCDS-B Home
• This Issue
• Next Article
Connected components of positive solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space
June  2019, 24(6): 2719-2743. doi: 10.3934/dcdsb.2018272

## Long time behavior of fractional impulsive stochastic differential equations with infinite delay

 1 School of Mathematics and Statistics, Xi’an Jiaotong University Xi’an 710049, China 2 Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, c/ Tarfia s/n, 41012 Sevilla, Spain

Received  March 2018 Revised  May 2018 Published  October 2018

Fund Project: This work has been supported by grant MTM2015-63723-P (MINECO/FEDER, EU) and Consejería de Innovación, Ciencia y Empresa (Junta de Andalucía) under grant 2010/FQM314, and Proyecto de Excelencia P12-FQM-1492

This paper is first devoted to the local and global existence of mild solutions for a class of fractional impulsive stochastic differential equations with infinite delay driven by both $\mathbb{K}$-valued Q-cylindrical Brownian motion and fractional Brownian motion with Hurst parameter $H∈(1/2,1)$. A general framework which provides an effective way to prove the continuous dependence of mild solutions on initial value is established under some appropriate assumptions. Furthermore, it is also proved the exponential decay to zero of solutions to fractional stochastic impulsive differential equations with infinite delay.

Citation: Jiaohui Xu, Tomás Caraballo. Long time behavior of fractional impulsive stochastic differential equations with infinite delay. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2719-2743. doi: 10.3934/dcdsb.2018272
##### References:

show all references

##### References:
 [1] Yong Ren, Xuejuan Jia, Lanying Hu. Exponential stability of solutions to impulsive stochastic differential equations driven by $G$-Brownian motion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2157-2169. doi: 10.3934/dcdsb.2015.20.2157 [2] María J. Garrido–Atienza, Kening Lu, Björn Schmalfuss. Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 473-493. doi: 10.3934/dcdsb.2010.14.473 [3] Ahmed Boudaoui, Tomás Caraballo, Abdelghani Ouahab. Stochastic differential equations with non-instantaneous impulses driven by a fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2521-2541. doi: 10.3934/dcdsb.2017084 [4] Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019213 [5] Litan Yan, Xiuwei Yin. Optimal error estimates for fractional stochastic partial differential equation with fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 615-635. doi: 10.3934/dcdsb.2018199 [6] Eduardo Cuesta. Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations. Conference Publications, 2007, 2007 (Special) : 277-285. doi: 10.3934/proc.2007.2007.277 [7] Matteo Bonforte, Yannick Sire, Juan Luis Vázquez. Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5725-5767. doi: 10.3934/dcds.2015.35.5725 [8] Yong Ren, Wensheng Yin, Dongjin Zhu. Exponential stability of SDEs driven by $G$-Brownian motion with delayed impulsive effects: average impulsive interval approach. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3347-3360. doi: 10.3934/dcdsb.2018248 [9] Miguel V. S. Frasson, Patricia H. Tacuri. Asymptotic behaviour of solutions to linear neutral delay differential equations with periodic coefficients. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1105-1117. doi: 10.3934/cpaa.2014.13.1105 [10] Chun Wang, Tian-Zhou Xu. Stability of the nonlinear fractional differential equations with the right-sided Riemann-Liouville fractional derivative. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 505-521. doi: 10.3934/dcdss.2017025 [11] Ilknur Koca. Numerical analysis of coupled fractional differential equations with Atangana-Baleanu fractional derivative. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 475-486. doi: 10.3934/dcdss.2019031 [12] Fahd Jarad, Sugumaran Harikrishnan, Kamal Shah, Kuppusamy Kanagarajan. Existence and stability results to a class of fractional random implicit differential equations involving a generalized Hilfer fractional derivative. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 723-739. doi: 10.3934/dcdss.2020040 [13] Baskar Sundaravadivoo. Controllability analysis of nonlinear fractional order differential systems with state delay and non-instantaneous impulsive effects. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020138 [14] Guolian Wang, Boling Guo. Stochastic Korteweg-de Vries equation driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5255-5272. doi: 10.3934/dcds.2015.35.5255 [15] Yong Xu, Rong Guo, Di Liu, Huiqing Zhang, Jinqiao Duan. Stochastic averaging principle for dynamical systems with fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1197-1212. doi: 10.3934/dcdsb.2014.19.1197 [16] Yong Xu, Bin Pei, Rong Guo. Stochastic averaging for slow-fast dynamical systems with fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2257-2267. doi: 10.3934/dcdsb.2015.20.2257 [17] Luis Caffarelli, Juan-Luis Vázquez. Asymptotic behaviour of a porous medium equation with fractional diffusion. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1393-1404. doi: 10.3934/dcds.2011.29.1393 [18] Tyrone E. Duncan. Some linear-quadratic stochastic differential games for equations in Hilbert spaces with fractional Brownian motions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5435-5445. doi: 10.3934/dcds.2015.35.5435 [19] Xiaowei Tang, Xilin Fu. New comparison principle with Razumikhin condition for impulsive infinite delay differential systems. Conference Publications, 2009, 2009 (Special) : 739-743. doi: 10.3934/proc.2009.2009.739 [20] Kolade M. Owolabi, Abdon Atangana. High-order solvers for space-fractional differential equations with Riesz derivative. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 567-590. doi: 10.3934/dcdss.2019037

2018 Impact Factor: 1.008